Specialization & Trade
A Producer Optimum
Overview of Economics
Production and Production Possibilities
The Production Function
Production refers to the conversion of inputs, the
factors of production, into desired output. A production function
for a particular good or service is often written as follows:
where Xi is the quantity produced of a
particular good or service
- L represents the quantity and
ability of labor input available to the
- K represents capital input, machinery,
transportation equipment, and other types of intermediate goods.
- M represents land,
natural resources and raw material inputs for production, and
- R represents entrepreneurship, organization and risk-taking.
A positive relationship exists among these inputs and
the output such that greater availability of any of these
factors will lead to a greater potential for producing
output. In addition, all factors are assumed to be
essential for production to take place.
The functional relationship f(.)
represents a certain level of technology and know how, that presently
exists, for conversion these inputs into output such that any
technological improvements can also lead to the production of greater levels of output.
Production in the Short Run
In order to better understand the technological nature of production,
we distinguish between short run production
relationships where only one factor input may vary (typically labor)
in quantity holding the other factors of production constant (i.e., capital and/or
materials) and the long run where all factors
of production may vary. The short run allows for the development of a simple two
variable model to understand the behavior between a single variable input
and the corresponding level of output. Thus we can write:
For example we could develop a short run model for agricultural production
where the output is measures as kilograms of grain and labor is the variable input.
The fixed factors of production include the following:
We might hypothesize the production relationship to be as follows:
- 1 plow
- 1 tractor.... capital
- 1 truck
- 1 acre of land
- 10 kilograms of seed grain
(Constant Marginal Productivity)
|0||0 kg|| -|
| :|| :||100|
In this example we find that each time we add one more unit of labor,
output increases by 100 kg. The third column MPL defines
this relationship. This column measures the marginal
productivity of labor -- a measure of the contribution of each
additional unit of labor input to the level of output. In this case, we have a
situation of constant marginal productivity which is unrealistic with
production in the short run. Constant marginal productivity implies that as
labor input increases, output always increases without bound -- a situation
difficult to imagine with limited capital and one acre of land.
A more realistic situation would be that of diminishing
marginal productivity where increasing quantities of a single
input lead to less and less additional output. This property is just
an acknowledgment that it is impossible to produce an infinite level
of output when some factors of production (machines or land) fixed in
quantity. Numerically, we can model diminishing marginal productivity
as follows: Table 2
(Diminishing Marginal Productivity)
|0||0 kg|| -|
In this case, additional labor input results in additional output. However,
the contribution of each additional unit of labor is less than previous units such
that the sixth unit of labor contributes nothing to output. With 5 or
6 workers, the available amount of land cannot support additional output.
A short run production relationship can be modeled in the diagram
below. In this example, labor is the variable factor input and land, capital,
and entrepreneurship are fixed in quantity. There is a positive relationship
between labor input and output levels, however, as additional labor in used, less and
less additional output is produced (click on the second button). The shape
of this production function is consistent with
the law of diminishing marginal productivity.Figure 1
in Labor Input
in Capital Input
Changes in the amount of capital or other fixed factors or in the level of
technology will lead to an upward shift in the production function (click on the
third button) such that a greater level of output may be produced with the same
amount of labor input.
Production Possibilities & Opportunity Costs
If we extend our model of production to two (or several) goods, we can
develop a more realistic notion of production relationships. In a world of
scarce resources, business firms producing different goods are competing
for the same pool of factor inputs. In the short run labor is available for
the production of one or a combination of goods. However, the desire to increase
production of one good 'X' will come at the expense of another good 'Y' as labor
or other resources are relallocated from the first good to the second.
In the table below, we can model this competition for resources between
two goods: Apples and Bread. In this example, an additional unit of labor
directed to bread production allows for producing 25 additional units of
bread (the marginal productivity of labor in bread production is constant).
Separately, additional units of labor applied to apple production allows for
the producing anywhere from 0 to 185 units of apples.
Production Possibilities: Bread and Apples
(7 units of Labor available)
Working with an assumption that
the amount of labor input is fixed in supply at 7 units, this labor must be shared
between bread and apple production. If we happen to be producing 185 units of apples
(point A), then all 7 units of labor are being used for this purpose.
The desire to produce bread requires a reallocation of labor from apple
production to bread production. In the table above, we can show this
as a movement from point A to point
B. We are able to produce 25 units of bread (0-25 units) but
at the expense producing 6 apples (185-179 units). These six apples represent the
opportunity cost of bread production (i.e., each unit of bread "costs" 6/25 unit of apples.
If we move from point B
to point C, the additional 25 units of bread (25-50), come at the
expense of 16 apples (179-163 units). Because of diminishing marginal productivity
in apple production, the production of 25 additional units of bread requires
that more and more apples are given up. Stated differently, we can say that the opportunity cost of
producing bread is increasing.
The diagram below summarizes the numbers in the above table.
Points on the blue curve -- the Production Possibilities
Frontier represent an efficient use of resources. Points within the
curve represent inefficient production levels -- resources and technology
allow for producing more of good X, good Y, or more of both.
Movement along the curve (use the scrollbar to see changes) imply that a tradeoff exists in production when resources are
scarce or fixed in supply. Finally points (combinations of the two goods)
beyond the frontier are unattainable with existing levels of technology and
The Marginal Rate of Transformation
The diagram below defines the slope of this same PPF at any given point.
This slope, is known as the Marginal Rate of Transformation
(MRT), is a measure of the ratio of
MRT = MPy /MPx =
Marginal Costx /Marginal Costy
This ratio measures the opportunity cost of using resources in
producing one good in terms of the alternative use of those resources used in the production of other goods.
Given the role of diminishing marginal productivity; as resources are allocated away from good Y towards good
X, the opportunity cost (|MRT|), of producing more of good X, increases.
given that: Marginal Costi = wage rate /MPi
If resources were to be allocated in the opposite direction, the same would be true -- the opportunity
cost of producing more of good Y would also increase in terms of foregone production of good X.
Opportunity Costs and Relative Prices
Suppose that we are producing at point D in the above diagrams. If we transfer one unit of
labor away from apple production to bread production, we must give up 30 units of apples and gain 25 units of bread.
Thus the opportunity "cost" of each unit of bread is 6/5 (1.20) of an apple. Bringing
relative prices into the picture, we might find
that the price of apples 'Papples' is $2.00 and the price of bread 'Pbread is $3.00.
PR = [Px / PY] = [Pbread / Papples] = 1.50.
(as a Terms of Trade), we find that we are willing to trade 1 unit of bread for $3.00 and with that $3.00, we could then acquire
1.5 units of apples. One unit of bread is "worth" 1.5 units of apples given these prices.
If we compare the relative price of bread to the opportunity cost of bread, we find that the value of
bread in terms of apples is greater than the opportunity cost of producing bread:
one loaf of bread is worth 1.50 apples
From a social point of view, we should be allocating more resources towards bread production and
away from apple production. With the reallocation of resources, the opportunity cost of bread production will rise.
This reallocation should continue until the following is true:
one loaf of bread costs 1.20 apples, or
Pbread / Papples > MRT
PR = MRT
Pbread / Papples = MCbread / MCapples
If the price of bread were to fall, say to $2.00 per unit (click on the Next button),
then the relative price of bread would
be equal to 1.0 (i.e., each unit of bread is exactly equal to the value of one unit of apples) and PR
this change, the value of bread, in terms of apples, is less than the opportunity costs of
producing bread. In this case we should allocate resources away from bread production (click on the Next button again). This reallocation
will cause the opportunity cost of bread to fall until it is just equal to the prevailing relative price.
(Click on the Next button again to repeat.)
This example above, demonstrates how the relationship between relative prices
(acting as signals about the "value" place on products by consumers)
and opportunity costs (that represent the underlying characteristics of
production technology) determine an efficient use of resources. As relative prices
change, resources will be reallocated to that good that is valued more highly taking into
account the opportunity costs of additional production of that good.
Concepts for Review:
- Diminishing Marginal Productivity
- Inefficient Production
- Long Run Production
- Marginal Productivity of Labor
- Marginal Rate of Transformation
- Opportunity Cost
- Production Function
- Production Possibilities
- Relative Prices
- Short Run Production
- Unattainable Output Combinations